Every positive rational number has eventually greedy best Egyptian underapproximations, with and without repeated denominators, and a Liouville example with unique greedy best underapproximations is constructed.
Asymptotic Analysis of Infinite Decompositions of a Unit Fraction into Unit Fractions
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Paul Erd\H{o}s posed a problem on the asymptotic estimation of decomposing 1 into a sum of infinitely many unit fractions in \cite{Erd80}. We point out that this problem can be solved in the same way as the finite case, as shown in \cite{Sou05}.
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Eventually greedy best Egyptian underapproximations of rational numbers via optimal control
Every positive rational number has eventually greedy best Egyptian underapproximations, with and without repeated denominators, and a Liouville example with unique greedy best underapproximations is constructed.